Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={2 ones, 1 three, 2 fours and 3 sixes}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (''one'' is the most likely and ''six'' is the least likely outcome).

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<center>

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{| class="wikitable" style="text-align:center; width:75%" border="1"

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|-

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| ''x'' || 1 || 2 || 3 || 4 || 5 || 6

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|-

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| ''P(X=x)'' || 0.286 || 0.238 || 0.19 || 0.143 || 0.095 || 0.048

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|}

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</center>

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: ''P(A)=?''

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Of course, we can compute this number exactly as:

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: <math>P(A) =</math>

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However, we can also find a pretty close empirically-driven estimate using the [[SOCR_EduMaterials_Activities_DiceSampleExperiment | SOCR Dice Sample Experiment]].

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For instance, running the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Dice Sample Experiment] 1,000 times with number of dice n=10, and the loading probabilities listed above, we get an output like the one shown below.

Now, we can actually count how many of these 1,000 trials generated the event ''A'' as an outcome. Then the relative proportion of these outcomes to 1,000 will give us a fairly accurate estimate of the exact probability we computed above

Multinomial experiments

A multinomial experiment is an experiment that has the following properties:

The experiment consists of k repeated trials.

Each trial has a discrete number of possible outcomes.

On any given trial, the probability that a particular outcome will occur is constant.

The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

Examples of Multinomial experiments

Suppose we have an urn containing 9 marbles. Two are red, three are green, and four are blue (2+3+4=9). We randomly select 5 marbles from the urn, with replacement. What is the probability (P(A)) of the event A={selecting 2 green marbles and 3 blue marbles}?

To solve this problem, we apply the multinomial formula. We know the following:

Synergies between Binomial and Multinomial processes/probabilities/coefficients

Example

Suppose we study N independent trials with results falling in one of k possible categories labeled 1,2,cdots,k. Let pi be the probability of a trial resulting in the ith category, where . Let Ni be the number of trials resulting in the ith category, where .

For instance, suppose we have 9 people arriving at a meeting according to the following information:

SOCR Multinomial Examples

Suppose we row 10 loaded hexagonal (6-face) dice 8 times and we are interested in the probability of observing the event A={2 ones, 1 three, 2 fours and 3 sixes}. Assume the dice are loaded to the small outcomes according to the following probabilities of the 6 outcomes (one is the most likely and six is the least likely outcome).

For instance, running the SOCR Dice Sample Experiment 1,000 times with number of dice n=10, and the loading probabilities listed above, we get an output like the one shown below.

Now, we can actually count how many of these 1,000 trials generated the event A as an outcome. Then the relative proportion of these outcomes to 1,000 will give us a fairly accurate estimate of the exact probability we computed above